Bike Sharing Company Assignment

Description:

BoomBikes Company -A bike-sharing system/provider is a service in which bikes are made available for shared use to individuals on a short term basis for a price or free. Many bike share systems allow people to borrow a bike from a "dock" which is usually computer-controlled wherein the user enters the payment information, and the system unlocks it. This bike can then be returned to another dock belonging to the same system.

Technical Objective:

The company wants to know:

1.Which variables are significant in predicting the demand for shared bikes.

2.How well those variables describe the bike demands

Business Goal:

You are required to model the demand for shared bikes with the available independent variables. It will be used by the management to understand how exactly the demands vary with different features. They can accordingly manipulate the business strategy to meet the demand levels and meet the customer's expectations. Further, the model will be a good way for management to understand the demand dynamics of a new market.

Steps:

1. Read and visualise the Dataset
2. Data preparation
3. Split the data set into train and test sets
4. Build Model with training dataset
5. Estimation
6. Inference
7. Model Prediction

Step1: Read and visualise the Dataset

a. Reading and Understanding of Data

# Import header files
import numpy as np
import pandas as pd
import missingno as msno
import matplotlib.pyplot as plt
%matplotlib inline
import seaborn as sns
import plotly
import plotly.graph_objects as go
from plotly.subplots import make_subplots

# Supress Warnings
import warnings
warnings.filterwarnings('ignore')

#load/read data 
Bike = pd.read_csv("day.csv", low_memory=False)

Bike.info()

<class 'pandas.core.frame.DataFrame'>
RangeIndex: 730 entries, 0 to 729
Data columns (total 16 columns):
 #   Column      Non-Null Count  Dtype  
---  ------      --------------  -----  
 0   instant     730 non-null    int64  
 1   dteday      730 non-null    object 
 2   season      730 non-null    int64  
 3   yr          730 non-null    int64  
 4   mnth        730 non-null    int64  
 5   holiday     730 non-null    int64  
 6   weekday     730 non-null    int64  
 7   workingday  730 non-null    int64  
 8   weathersit  730 non-null    int64  
 9   temp        730 non-null    float64
 10  atemp       730 non-null    float64
 11  hum         730 non-null    float64
 12  windspeed   730 non-null    float64
 13  casual      730 non-null    int64  
 14  registered  730 non-null    int64  
 15  cnt         730 non-null    int64  
dtypes: float64(4), int64(11), object(1)
memory usage: 91.4+ KB

Insight:

All are numberic variables and no missing data

Dataset characteristics

day.csv have the following fields:

• instant: record index
• dteday : date
• season : season (1:spring, 2:summer, 3:fall, 4:winter)
• yr : year (0: 2018, 1:2019)m
• mnth : month ( 1 to 12)
• holiday : weather day is a holiday or not (extracted from http://dchr.dc.gov/page/holiday-schedule)
• weekday : day of the week
• workingday : if day is neither weekend nor holiday is 1, otherwise is 0.
• weathersit :
  • 1: Clear, Few clouds, Partly cloudy, Partly cloudy
  • 2: Mist + Cloudy, Mist + Broken clouds, Mist + Few clouds, Mist
  • 3: Light Snow, Light Rain + Thunderstorm + Scattered clouds, Light Rain + Scattered clouds
  • 4: Heavy Rain + Ice Pallets + Thunderstorm + Mist, Snow + Fog
• temp : temperature in Celsius
• atemp: feeling temperature in Celsius
• hum: humidity
• windspeed: wind speed
• casual: count of casual users
• registered: count of registered users
• cnt: count of total rental bikes including both casual and registered

Insight:

1. We need predit the cnt column - count of total rental bikes so,our traget variable is 'cnt'

cnt is an continous variable, and there are multiple features available - Choosing multiple linear regression model for the prediction

#check the size
Bike.shape

(730, 16)

Insight:

There are 730 rows and 16 columns

#check the head of dataset
Bike.head()

	instant	dteday	season	yr	mnth	holiday	weekday	workingday	weathersit	temp	atemp	hum	windspeed	casual	registered	cnt
0	1	01-01-2018	1	0	1	0	6	0	2	14.110847	18.18125	80.5833	10.749882	331	654	985
1	2	02-01-2018	1	0	1	0	0	0	2	14.902598	17.68695	69.6087	16.652113	131	670	801
2	3	03-01-2018	1	0	1	0	1	1	1	8.050924	9.47025	43.7273	16.636703	120	1229	1349
3	4	04-01-2018	1	0	1	0	2	1	1	8.200000	10.60610	59.0435	10.739832	108	1454	1562
4	5	05-01-2018	1	0	1	0	3	1	1	9.305237	11.46350	43.6957	12.522300	82	1518	1600

#check the tail of dataset
Bike.tail()

	instant	dteday	season	yr	mnth	holiday	weekday	workingday	weathersit	temp	atemp	hum	windspeed	casual	registered	cnt
725	726	27-12-2019	1	1	12	0	4	1	2	10.420847	11.33210	65.2917	23.458911	247	1867	2114
726	727	28-12-2019	1	1	12	0	5	1	2	10.386653	12.75230	59.0000	10.416557	644	2451	3095
727	728	29-12-2019	1	1	12	0	6	0	2	10.386653	12.12000	75.2917	8.333661	159	1182	1341
728	729	30-12-2019	1	1	12	0	0	0	1	10.489153	11.58500	48.3333	23.500518	364	1432	1796
729	730	31-12-2019	1	1	12	0	1	1	2	8.849153	11.17435	57.7500	10.374682	439	2290	2729

Insight:

Instant column is the index value - which is not needed

Bike.describe()

	instant	season	yr	mnth	holiday	weekday	workingday	weathersit	temp	atemp	hum	windspeed	casual	registered	cnt
count	730.000000	730.000000	730.000000	730.000000	730.000000	730.000000	730.000000	730.000000	730.000000	730.000000	730.000000	730.000000	730.000000	730.000000	730.000000
mean	365.500000	2.498630	0.500000	6.526027	0.028767	2.997260	0.683562	1.394521	20.319259	23.726322	62.765175	12.763620	849.249315	3658.757534	4508.006849
std	210.877136	1.110184	0.500343	3.450215	0.167266	2.006161	0.465405	0.544807	7.506729	8.150308	14.237589	5.195841	686.479875	1559.758728	1936.011647
min	1.000000	1.000000	0.000000	1.000000	0.000000	0.000000	0.000000	1.000000	2.424346	3.953480	0.000000	1.500244	2.000000	20.000000	22.000000
25%	183.250000	2.000000	0.000000	4.000000	0.000000	1.000000	0.000000	1.000000	13.811885	16.889713	52.000000	9.041650	316.250000	2502.250000	3169.750000
50%	365.500000	3.000000	0.500000	7.000000	0.000000	3.000000	1.000000	1.000000	20.465826	24.368225	62.625000	12.125325	717.000000	3664.500000	4548.500000
75%	547.750000	3.000000	1.000000	10.000000	0.000000	5.000000	1.000000	2.000000	26.880615	30.445775	72.989575	15.625589	1096.500000	4783.250000	5966.000000
max	730.000000	4.000000	1.000000	12.000000	1.000000	6.000000	1.000000	3.000000	35.328347	42.044800	97.250000	34.000021	3410.000000	6946.000000	8714.000000

Insight:

1.There are no missing data.

2.All values are numeric.

3.There are 730 entries and 16 columns.

4.Values are in different ranges.

b. Data Visualization

Numeric Data Analysis

All the data in the table Numeric except Date

# lets check the correlation to see which variables are highly correlated

plt.figure(figsize = (15,10))
sns.heatmap(Bike.corr(), annot = True, cmap="YlGnBu")
plt.show()

Insight:

High correlation observed for cnt with temp and atemp ( ignoring instant , casual,registered)

Pairplot for Visualization

sns.pairplot(Bike)
plt.show()

Insight:

1. Right side plots looks highly correlated.
2. cnt is having linearity with temp, atemp ( ignoring casual,registered)
3. We can observe that few columns are highly correlated from both Correlation and pairplot

c. Data Cleanup:

Drop unwanted columns

1. instant - Index column is not needed

2. dteday - date column is not needed as the data is available in Different columns

3. casual + registered = cnt, these two columns not needed and cnt need to be predicted

The model should be built taking this 'cnt' as the target variable.

#Drop column list
datalist = ['instant','dteday', 'casual', 'registered']
#Drop columns
Bike = Bike.drop(datalist, axis = 1)

d.Data Analysis

plt.figure(figsize = (20,12))
plt.subplot(2,2,1)
sns.boxplot(x='weathersit', y = 'cnt', data = Bike)
plt.subplot(2,2,2)
sns.boxplot(x='yr', y = 'cnt', data = Bike)
plt.subplot(2,2,3)
sns.boxplot(x='season', y = 'cnt', data = Bike)
plt.subplot(2,2,4)
sns.boxplot(x='mnth', y = 'cnt', data = Bike)
plt.show()

Insight:

1. Weathersit is having only 3 values - no bike is used during 4: Heavy Rain + Ice Pallets + Thunderstorm + Mist, Snow + Fog
2. All the above columns ar having some impact on the cnt variable

# as there are two years lets plot and understand in detail
# Multivariante analysis  - cnt, year, month
plt.figure(figsize=(10,5))
sns.boxplot(x='mnth', y = 'cnt', hue = 'yr', data = Bike)
plt.show()

Insight:

1. Usage of bikes are high in year 2019 (0-2018, 1-2019)
2. Usage is more from 3rd month

plt.figure(figsize = (20,6))
plt.subplot(1,3,1)
sns.boxplot(x='holiday', y = 'cnt', data = Bike)
plt.subplot(1,3,2)
sns.boxplot(x='weekday', y = 'cnt', data = Bike)
plt.subplot(1,3,3)
sns.boxplot(x='workingday', y = 'cnt', data = Bike)

plt.show()

Insight:

1. Even though above columns are numeric we need to treat them as Categorical variables
2. Create dummy variables

datalist = ['temp', 'atemp', 'hum', 'windspeed', 'cnt']
sns.pairplot(Bike[datalist])
plt.show()

# Check correlation for the same variables
plt.figure(figsize = (15,10))
sns.heatmap(Bike[datalist].corr(), annot = True, cmap="YlGnBu")
plt.show()

Insights:

1. High correlation between temp and atemp columns -0.99 ,we can consider one column.
2. Correlation of temp and atemp is same with the cnt(target variable) - 0.63
3. for Bike ride feel temperater will be better to choose than the actual temperature - analyse with linear regression
4. atemp column is having high correlation with the target variable cnt

Step2 : Data Preparation

a. Creating the dummy variables for categorical variables using One hot encoding

Categorical Variables :

• season : season (1:spring, 2:summer, 3:fall, 4:winter)
• yr : year (0: 2018, 1:2019)
• mnth : month ( 1 to 12)
• holiday : weather day is a holiday or not
• weekday : day of the week - 0,6 are non working days(weekend)
• workingday : if day is neither weekend nor holiday is 1, otherwise is 0.
• weathersit : 1: Clear, Few clouds, Partly cloudy, Partly cloudy 2: Mist + Cloudy, Mist + Broken clouds, Mist + Few clouds, Mist 3: Light Snow, Light Rain + Thunderstorm + Scattered clouds, Light Rain + Scattered clouds 4: Heavy Rain + Ice Pallets + Thunderstorm + Mist, Snow + Fog

Note: No need to create dummy variables for holiday and working day as it is similar to yes and no and holding 0 and 1 value

seasons = pd.get_dummies(Bike.season, drop_first = True)

seasons

	2	3	4
0	0	0	0
1	0	0	0
2	0	0	0
3	0	0	0
4	0	0	0
...	...	...	...
725	0	0	0
726	0	0	0
727	0	0	0
728	0	0	0
729	0	0	0

730 rows × 3 columns

Insight

1. As all categorical variable are having value in range [0-12] , it is needed to change the column names

#renaming column names for seasons table
seasons.rename(columns={2:'summer', 3:'fall', 4:'winter'}, inplace = True)
seasons

	summer	fall	winter
0	0	0	0
1	0	0	0
2	0	0	0
3	0	0	0
4	0	0	0
...	...	...	...
725	0	0	0
726	0	0	0
727	0	0	0
728	0	0	0
729	0	0	0

730 rows × 3 columns

# Dummy variables for weekday (sunday is dropped) and renaming columns name
weekday = pd.get_dummies(Bike.weekday,drop_first = True)
weekday.rename(columns={1:'mon', 2:'tue', 3:'wed', 4:'thu', 5:'fri', 6:'sat'}, inplace = True)
weekday

	mon	tue	wed	thu	fri	sat
0	0	0	0	0	0	1
1	0	0	0	0	0	0
2	1	0	0	0	0	0
3	0	1	0	0	0	0
4	0	0	1	0	0	0
...	...	...	...	...	...	...
725	0	0	0	1	0	0
726	0	0	0	0	1	0
727	0	0	0	0	0	1
728	0	0	0	0	0	0
729	1	0	0	0	0	0

730 rows × 6 columns

# Dummy variables for weather (clear weather is dropped) and renaming columns name
weather = pd.get_dummies(Bike.weathersit,drop_first = True)
weather.rename(columns={1:'w_clear', 2:'w_mist', 3:'w_light'}, inplace = True)
weather

	w_mist	w_light
0	1	0
1	1	0
2	0	0
3	0	0
4	0	0
...	...	...
725	1	0
726	1	0
727	1	0
728	0	0
729	1	0

730 rows × 2 columns

#Dummy variables for year 
year = pd.get_dummies(Bike.yr, drop_first = True)
year.rename(columns={1:'yr_2019'}, inplace = True)
year

	yr_2019
0	0
1	0
2	0
3	0
4	0
...	...
725	1
726	1
727	1
728	1
729	1

730 rows × 1 columns

# Dummy variables for weather (jan is dropped) and add prefix as month to the columns
month = pd.get_dummies(Bike.mnth,drop_first= True)
month = month.add_prefix("month_")
month

	month_2	month_3	month_4	month_5	month_6	month_7	month_8	month_9	month_10	month_11	month_12
0	0	0	0	0	0	0	0	0	0	0	0
1	0	0	0	0	0	0	0	0	0	0	0
2	0	0	0	0	0	0	0	0	0	0	0
3	0	0	0	0	0	0	0	0	0	0	0
4	0	0	0	0	0	0	0	0	0	0	0
...	...	...	...	...	...	...	...	...	...	...	...
725	0	0	0	0	0	0	0	0	0	0	1
726	0	0	0	0	0	0	0	0	0	0	1
727	0	0	0	0	0	0	0	0	0	0	1
728	0	0	0	0	0	0	0	0	0	0	1
729	0	0	0	0	0	0	0	0	0	0	1

730 rows × 11 columns

# Creating list for catgorical columns
catlist = ['season', 'yr', 'mnth', 'weekday', 'weathersit']

# Drop the categorical columns from the Bike table.
Bike_New = Bike.drop(catlist, axis = 1)

#Creating table for all dummy variables(categorial variables)
cat_concat = pd.concat([seasons, year, month, weekday, weather], axis = 1)

# Concat the dummy variable table with the Bike_New (main)
Bike_New = pd.concat([Bike_New,cat_concat], axis = 1)

Bike_New.shape

(730, 30)

Insight:

There are 30 columns after coverting the categorical variables into dummy

Bike_New.head()

	holiday	workingday	temp	atemp	hum	windspeed	cnt	summer	fall	winter	...	month_11	month_12	mon	tue	wed	thu	fri	sat	w_mist	w_light
0	0	0	14.110847	18.18125	80.5833	10.749882	985	0	0	0	...	0	0	0	0	0	0	0	1	1	0
1	0	0	14.902598	17.68695	69.6087	16.652113	801	0	0	0	...	0	0	0	0	0	0	0	0	1	0
2	0	1	8.050924	9.47025	43.7273	16.636703	1349	0	0	0	...	0	0	1	0	0	0	0	0	0	0
3	0	1	8.200000	10.60610	59.0435	10.739832	1562	0	0	0	...	0	0	0	1	0	0	0	0	0	0
4	0	1	9.305237	11.46350	43.6957	12.522300	1600	0	0	0	...	0	0	0	0	1	0	0	0	0	0

5 rows × 30 columns

Step 3: Splitting the dataset into Training and Test sets

a. The First basis step in regression is performing Training - Test split

#import sklearn for splitting
from sklearn.model_selection import train_test_split

np.random.seed(0)
# Data split into 70:30 ratio of train and test 
df_train, df_test = train_test_split(Bike_New,train_size = 0.7, test_size = 0.3, random_state =100 )

b.Rescaling the features

as we saw the data was in different ranges -Scaling will not impact any parameters except co-efficient and this will help ease of interpretation of linear model

We are using MinMax scaling

# Import 
from sklearn.preprocessing import MinMaxScaler

scaler = MinMaxScaler()

# Apply scaler() to all the columns except dummy and 0, 1 variables
num_vars = ['temp', 'atemp', 'hum', 'windspeed', 'cnt']

df_train[num_vars] = scaler.fit_transform(df_train[num_vars])

df_train.head()

	holiday	workingday	temp	atemp	hum	windspeed	cnt	summer	fall	winter	...	month_11	month_12	mon	tue	wed	thu	fri	sat	w_mist	w_light
653	0	1	0.509887	0.501133	0.575354	0.300794	0.864243	0	0	1	...	0	0	0	1	0	0	0	0	0	0
576	0	1	0.815169	0.766351	0.725633	0.264686	0.827658	0	1	0	...	0	0	0	1	0	0	0	0	0	0
426	0	0	0.442393	0.438975	0.640189	0.255342	0.465255	0	0	0	...	0	0	0	0	0	0	0	1	1	0
728	0	0	0.245101	0.200348	0.498067	0.663106	0.204096	0	0	0	...	0	1	0	0	0	0	0	0	0	0
482	0	0	0.395666	0.391735	0.504508	0.188475	0.482973	1	0	0	...	0	0	0	0	0	0	0	1	1	0

5 rows × 30 columns

df_train.describe()

	holiday	workingday	temp	atemp	hum	windspeed	cnt	summer	fall	winter	...	month_11	month_12	mon	tue	wed	thu	fri	sat	w_mist	w_light
count	510.000000	510.000000	510.000000	510.000000	510.000000	510.000000	510.000000	510.000000	510.000000	510.00000	...	510.000000	510.000000	510.000000	510.000000	510.000000	510.000000	510.000000	510.000000	510.000000	510.000000
mean	0.025490	0.676471	0.537262	0.512989	0.650369	0.320768	0.513620	0.245098	0.262745	0.24902	...	0.086275	0.084314	0.150980	0.131373	0.158824	0.133333	0.127451	0.154902	0.343137	0.029412
std	0.157763	0.468282	0.225844	0.212385	0.145882	0.169797	0.224593	0.430568	0.440557	0.43287	...	0.281045	0.278131	0.358381	0.338139	0.365870	0.340268	0.333805	0.362166	0.475223	0.169124
min	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.00000	...	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000
25%	0.000000	0.000000	0.339853	0.332086	0.538643	0.199179	0.356420	0.000000	0.000000	0.00000	...	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000
50%	0.000000	1.000000	0.540519	0.526811	0.653714	0.296763	0.518638	0.000000	0.000000	0.00000	...	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000
75%	0.000000	1.000000	0.735215	0.688457	0.754830	0.414447	0.684710	0.000000	1.000000	0.00000	...	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	1.000000	0.000000
max	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1.00000	...	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000

8 rows × 30 columns

# Lets check the correlation
plt.figure(figsize = (30,30))
sns.heatmap(df_train.corr(), annot = True, cmap="YlGnBu")
plt.show()

Insight :

1. Highest corrlation observed with temp and atemp vs cnt

# Pair plot with the temp and cnt to check the relationship
plt.figure(figsize = [6,6])
plt.scatter(df_train.cnt,df_train.temp)
plt.show

<function matplotlib.pyplot.show(close=None, block=None)>

Insight:

1. Scatter plot distribution looks linear with the cnt and temp columns

c.Dividing into X and y sets for the model building

y_train = df_train.pop('cnt')
X_train = df_train

X_train.head()

	holiday	workingday	temp	atemp	hum	windspeed	summer	fall	winter	yr_2019	...	month_11	month_12	mon	tue	wed	thu	fri	sat	w_mist	w_light
653	0	1	0.509887	0.501133	0.575354	0.300794	0	0	1	1	...	0	0	0	1	0	0	0	0	0	0
576	0	1	0.815169	0.766351	0.725633	0.264686	0	1	0	1	...	0	0	0	1	0	0	0	0	0	0
426	0	0	0.442393	0.438975	0.640189	0.255342	0	0	0	1	...	0	0	0	0	0	0	0	1	1	0
728	0	0	0.245101	0.200348	0.498067	0.663106	0	0	0	1	...	0	1	0	0	0	0	0	0	0	0
482	0	0	0.395666	0.391735	0.504508	0.188475	1	0	0	1	...	0	0	0	0	0	0	0	1	1	0

5 rows × 29 columns

Insight:

1. After removing cnt columns , there are 29 independent variables (columns)

Step 4: Building a linear model using training data

Fit a regression line through the training data using statmodels + add constant using add_constant(X)

Create first fitting model

import statsmodels.api as sm

# add constant
X_train_lm = sm.add_constant(X_train)

#Create first fitting model
lr = sm.OLS(y_train,X_train_lm).fit()

lr.params

const         0.175618
holiday      -0.042394
workingday    0.043879
temp          0.401322
atemp         0.050628
hum          -0.151812
windspeed    -0.184388
summer        0.086796
fall          0.048580
winter        0.153968
yr_2019       0.232208
month_2       0.030389
month_3       0.063853
month_4       0.062565
month_5       0.087257
month_6       0.060862
month_7       0.023289
month_8       0.078641
month_9       0.144371
month_10      0.070260
month_11      0.020783
month_12      0.016994
mon          -0.009946
tue          -0.007753
wed           0.005823
thu           0.001806
fri           0.011556
sat           0.054533
w_mist       -0.061030
w_light      -0.256697
dtype: float64

# check the summary
print(lr.summary())

                            OLS Regression Results                            
==============================================================================
Dep. Variable:                    cnt   R-squared:                       0.853
Model:                            OLS   Adj. R-squared:                  0.845
Method:                 Least Squares   F-statistic:                     99.96
Date:                Sun, 27 Aug 2023   Prob (F-statistic):          8.42e-181
Time:                        16:05:17   Log-Likelihood:                 528.03
No. Observations:                 510   AIC:                            -998.1
Df Residuals:                     481   BIC:                            -875.3
Df Model:                          28                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
const          0.1756      0.030      5.777      0.000       0.116       0.235
holiday       -0.0424      0.024     -1.793      0.074      -0.089       0.004
workingday     0.0439      0.009      4.689      0.000       0.025       0.062
temp           0.4013      0.142      2.821      0.005       0.122       0.681
atemp          0.0506      0.138      0.366      0.714      -0.221       0.322
hum           -0.1518      0.039     -3.940      0.000      -0.228      -0.076
windspeed     -0.1844      0.026     -7.003      0.000      -0.236      -0.133
summer         0.0868      0.024      3.679      0.000       0.040       0.133
fall           0.0486      0.030      1.618      0.106      -0.010       0.108
winter         0.1540      0.026      5.932      0.000       0.103       0.205
yr_2019        0.2322      0.008     28.792      0.000       0.216       0.248
month_2        0.0304      0.021      1.474      0.141      -0.010       0.071
month_3        0.0639      0.022      2.857      0.004       0.020       0.108
month_4        0.0626      0.034      1.864      0.063      -0.003       0.129
month_5        0.0873      0.036      2.412      0.016       0.016       0.158
month_6        0.0609      0.039      1.556      0.120      -0.016       0.138
month_7        0.0233      0.044      0.529      0.597      -0.063       0.110
month_8        0.0786      0.042      1.873      0.062      -0.004       0.161
month_9        0.1444      0.037      3.853      0.000       0.071       0.218
month_10       0.0703      0.034      2.041      0.042       0.003       0.138
month_11       0.0208      0.033      0.633      0.527      -0.044       0.085
month_12       0.0170      0.027      0.641      0.522      -0.035       0.069
mon           -0.0099      0.010     -1.023      0.307      -0.029       0.009
tue           -0.0078      0.011     -0.695      0.488      -0.030       0.014
wed            0.0058      0.011      0.554      0.580      -0.015       0.026
thu            0.0018      0.011      0.165      0.869      -0.020       0.023
fri            0.0116      0.011      1.031      0.303      -0.010       0.034
sat            0.0545      0.015      3.757      0.000       0.026       0.083
w_mist        -0.0610      0.010     -5.845      0.000      -0.082      -0.041
w_light       -0.2567      0.026     -9.712      0.000      -0.309      -0.205
==============================================================================
Omnibus:                       85.143   Durbin-Watson:                   2.041
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              237.880
Skew:                          -0.809   Prob(JB):                     2.21e-52
Kurtosis:                       5.929   Cond. No.                     1.22e+16
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The smallest eigenvalue is 1.17e-29. This might indicate that there are
strong multicollinearity problems or that the design matrix is singular.

Insight:

1. $R^2$ and adj.$R^2$ values are desent - 0.84
2. P-values for few features are high

Checking VIF (Variance Inflation Factor)

VIF gives a basic quantitative idea about how much the feature variables are correllated with each other and it is important parameter in the linear regression

VIF = $ 1 / (1-R^2) $

#import the VIF lib
from statsmodels.stats.outliers_influence import variance_inflation_factor

#create dataframe for all the features and their VIF's
vif = pd.DataFrame()
vif['Features'] = X_train.columns
vif['VIF'] = [variance_inflation_factor(X_train.values, i) for i in range(X_train.shape[1])]
vif['VIF'] = round(vif['VIF'],2)
vif = vif.sort_values(by = 'VIF' , ascending = False)
vif

	Features	VIF
0	holiday	inf
25	fri	inf
24	thu	inf
23	wed	inf
22	tue	inf
21	mon	inf
1	workingday	inf
2	temp	447.70
3	atemp	383.54
4	hum	20.79
7	fall	15.42
16	month_8	11.01
8	winter	10.93
15	month_7	9.62
6	summer	8.88
14	month_6	7.35
17	month_9	7.34
13	month_5	7.16
18	month_10	6.64
19	month_11	5.99
12	month_4	5.64
5	windspeed	4.71
20	month_12	3.78
11	month_3	3.06
27	w_mist	2.21
9	yr_2019	2.09
26	sat	1.93
10	month_2	1.71
28	w_light	1.23

Insight:

1. For few features VIF value is infinite.

2. $ VIF_i = \frac{1}{1 - {R_i}^2} $,VIF will become infinite when $ {R_i}^2 $ is 1

Which means that the model is able to explain 100% or 1.0 of the variance which indicates the perfect fit

1. If the VIF is high(>10) and the variable should be elimated which is causing perfect multicollinearity

Building model with the mixed approach

RFE - Recursive Feature Elimination

#import RFE and linear regression 
from sklearn.feature_selection import RFE
from sklearn.linear_model import LinearRegression

# Running RFE with the output feature = 20
lm = LinearRegression()
lm.fit(X_train,y_train)

rfe = RFE(lm, n_features_to_select = 20)
rfe = rfe.fit(X_train,y_train)

list(zip(X_train.columns, rfe.support_, rfe.ranking_))

[('holiday', True, 1),
 ('workingday', True, 1),
 ('temp', True, 1),
 ('atemp', True, 1),
 ('hum', True, 1),
 ('windspeed', True, 1),
 ('summer', True, 1),
 ('fall', True, 1),
 ('winter', True, 1),
 ('yr_2019', True, 1),
 ('month_2', False, 2),
 ('month_3', True, 1),
 ('month_4', True, 1),
 ('month_5', True, 1),
 ('month_6', True, 1),
 ('month_7', False, 3),
 ('month_8', True, 1),
 ('month_9', True, 1),
 ('month_10', True, 1),
 ('month_11', False, 4),
 ('month_12', False, 5),
 ('mon', False, 6),
 ('tue', False, 7),
 ('wed', False, 9),
 ('thu', False, 10),
 ('fri', False, 8),
 ('sat', True, 1),
 ('w_mist', True, 1),
 ('w_light', True, 1)]

X_train.columns[~rfe.support_]

Index(['month_2', 'month_7', 'month_11', 'month_12', 'mon', 'tue', 'wed',
       'thu', 'fri'],
      dtype='object')

Insight

1. above is the list of the columns eliminated by RFE

# selected list
col_list = X_train.columns[rfe.support_]
col_list

Index(['holiday', 'workingday', 'temp', 'atemp', 'hum', 'windspeed', 'summer',
       'fall', 'winter', 'yr_2019', 'month_3', 'month_4', 'month_5', 'month_6',
       'month_8', 'month_9', 'month_10', 'sat', 'w_mist', 'w_light'],
      dtype='object')

# Create dataframe for the selction list
X_train_rfe = X_train[col_list]
X_train_rfe

	holiday	workingday	temp	atemp	hum	windspeed	summer	fall	winter	yr_2019	month_3	month_4	month_5	month_6	month_8	month_9	month_10	sat	w_mist	w_light
653	0	1	0.509887	0.501133	0.575354	0.300794	0	0	1	1	0	0	0	0	0	0	1	0	0	0
576	0	1	0.815169	0.766351	0.725633	0.264686	0	1	0	1	0	0	0	0	0	0	0	0	0	0
426	0	0	0.442393	0.438975	0.640189	0.255342	0	0	0	1	1	0	0	0	0	0	0	1	1	0
728	0	0	0.245101	0.200348	0.498067	0.663106	0	0	0	1	0	0	0	0	0	0	0	0	0	0
482	0	0	0.395666	0.391735	0.504508	0.188475	1	0	0	1	0	1	0	0	0	0	0	1	1	0
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
526	0	1	0.824514	0.762183	0.605840	0.355596	1	0	0	1	0	0	0	1	0	0	0	0	1	0
578	0	1	0.863973	0.824359	0.679690	0.187140	0	1	0	1	0	0	0	0	1	0	0	0	0	0
53	0	1	0.202618	0.218747	0.435939	0.111379	0	0	0	0	0	0	0	0	0	0	0	0	0	0
350	0	0	0.248216	0.223544	0.577930	0.431816	0	0	1	0	0	0	0	0	0	0	0	1	1	0
79	0	1	0.462664	0.434043	0.759870	0.529881	1	0	0	0	1	0	0	0	0	0	0	0	1	0

510 rows × 20 columns

Insight:

1. we can see there are 20 columns (as specified in RFE) and 510 row now

Create fitting model -1

# add constant
X_train_rfe = sm.add_constant(X_train_rfe)

# Create fitting model 
lm = sm.OLS(y_train, X_train_rfe).fit()

#check summary
print(lm.summary())

                            OLS Regression Results                            
==============================================================================
Dep. Variable:                    cnt   R-squared:                       0.852
Model:                            OLS   Adj. R-squared:                  0.846
Method:                 Least Squares   F-statistic:                     140.5
Date:                Sun, 27 Aug 2023   Prob (F-statistic):          4.51e-188
Time:                        16:05:18   Log-Likelihood:                 525.37
No. Observations:                 510   AIC:                            -1009.
Df Residuals:                     489   BIC:                            -919.8
Df Model:                          20                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
const          0.1905      0.029      6.556      0.000       0.133       0.248
holiday       -0.0490      0.027     -1.829      0.068      -0.102       0.004
workingday     0.0436      0.011      3.801      0.000       0.021       0.066
temp           0.4339      0.136      3.190      0.002       0.167       0.701
atemp          0.0322      0.137      0.236      0.814      -0.236       0.301
hum           -0.1592      0.038     -4.220      0.000      -0.233      -0.085
windspeed     -0.1826      0.026     -7.027      0.000      -0.234      -0.132
summer         0.0871      0.022      4.041      0.000       0.045       0.129
fall           0.0489      0.023      2.126      0.034       0.004       0.094
winter         0.1574      0.014     11.110      0.000       0.130       0.185
yr_2019        0.2310      0.008     28.926      0.000       0.215       0.247
month_3        0.0472      0.017      2.816      0.005       0.014       0.080
month_4        0.0440      0.026      1.717      0.087      -0.006       0.094
month_5        0.0679      0.026      2.606      0.009       0.017       0.119
month_6        0.0390      0.022      1.737      0.083      -0.005       0.083
month_8        0.0569      0.018      3.155      0.002       0.021       0.092
month_9        0.1240      0.017      7.144      0.000       0.090       0.158
month_10       0.0487      0.017      2.819      0.005       0.015       0.083
sat            0.0534      0.014      3.696      0.000       0.025       0.082
w_mist        -0.0598      0.010     -5.791      0.000      -0.080      -0.040
w_light       -0.2533      0.026     -9.686      0.000      -0.305      -0.202
==============================================================================
Omnibus:                       82.801   Durbin-Watson:                   2.030
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              233.627
Skew:                          -0.784   Prob(JB):                     1.86e-51
Kurtosis:                       5.922   Cond. No.                         88.7
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

Insight

1. $ R^2 and Adj. R^2$ values are fine (0.84)
2. P-value is high for atemp

#create dataframe for all the features and their VIF's
vif = pd.DataFrame()
vif['Features'] = X_train_rfe.columns
vif['VIF'] = [variance_inflation_factor(X_train_rfe.values, i) for i in range(X_train_rfe.shape[1])]
vif['VIF'] = round(vif['VIF'],2)
vif = vif.sort_values(by = 'VIF' , ascending = False)
vif

	Features	VIF
3	temp	61.71
0	const	55.37
4	atemp	55.03
8	fall	6.71
7	summer	5.63
13	month_5	3.43
12	month_4	3.11
9	winter	2.46
14	month_6	2.28
5	hum	1.98
2	workingday	1.89
15	month_8	1.85
18	sat	1.79
11	month_3	1.63
19	w_mist	1.58
17	month_10	1.54
16	month_9	1.46
20	w_light	1.28
6	windspeed	1.27
1	holiday	1.17
10	yr_2019	1.04

Insight:

1. temp is having high VIF : 61.71 and low p value :0.002
2. atemp is having high VIF :55.03 and high p value :0.816 - will drop atemp column

#dropping atemp column
X_train_new = X_train_rfe.drop(['atemp'], axis = 1)

Rebuilding model without atemp -2

# add constant
X_train_new = sm.add_constant(X_train_new)

# Create fitting model 
lm = sm.OLS(y_train, X_train_new).fit()

#check summary
print(lm.summary())

                            OLS Regression Results                            
==============================================================================
Dep. Variable:                    cnt   R-squared:                       0.852
Model:                            OLS   Adj. R-squared:                  0.846
Method:                 Least Squares   F-statistic:                     148.2
Date:                Sun, 27 Aug 2023   Prob (F-statistic):          3.75e-189
Time:                        16:05:18   Log-Likelihood:                 525.34
No. Observations:                 510   AIC:                            -1011.
Df Residuals:                     490   BIC:                            -926.0
Df Model:                          19                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
const          0.1908      0.029      6.576      0.000       0.134       0.248
holiday       -0.0493      0.027     -1.843      0.066      -0.102       0.003
workingday     0.0436      0.011      3.804      0.000       0.021       0.066
temp           0.4648      0.037     12.674      0.000       0.393       0.537
hum           -0.1587      0.038     -4.217      0.000      -0.233      -0.085
windspeed     -0.1839      0.025     -7.240      0.000      -0.234      -0.134
summer         0.0871      0.022      4.043      0.000       0.045       0.129
fall           0.0484      0.023      2.115      0.035       0.003       0.093
winter         0.1575      0.014     11.139      0.000       0.130       0.185
yr_2019        0.2310      0.008     28.956      0.000       0.215       0.247
month_3        0.0473      0.017      2.823      0.005       0.014       0.080
month_4        0.0443      0.026      1.734      0.084      -0.006       0.094
month_5        0.0678      0.026      2.605      0.009       0.017       0.119
month_6        0.0387      0.022      1.728      0.085      -0.005       0.083
month_8        0.0564      0.018      3.152      0.002       0.021       0.092
month_9        0.1240      0.017      7.150      0.000       0.090       0.158
month_10       0.0487      0.017      2.822      0.005       0.015       0.083
sat            0.0534      0.014      3.699      0.000       0.025       0.082
w_mist        -0.0599      0.010     -5.804      0.000      -0.080      -0.040
w_light       -0.2538      0.026     -9.744      0.000      -0.305      -0.203
==============================================================================
Omnibus:                       82.440   Durbin-Watson:                   2.030
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              232.258
Skew:                          -0.781   Prob(JB):                     3.68e-51
Kurtosis:                       5.914   Cond. No.                         22.1
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

Insight

1. $ R^2 and Adj. R^2$ values are fine (0.85, 0.84)

#create dataframe for all the features and their VIF's
vif = pd.DataFrame()
vif['Features'] = X_train_new.columns
vif['VIF'] = [variance_inflation_factor(X_train_new.values, i) for i in range(X_train_new.shape[1])]
vif['VIF'] = round(vif['VIF'],2)
vif = vif.sort_values(by = 'VIF' , ascending = False)
vif

	Features	VIF
0	const	55.28
7	fall	6.65
6	summer	5.63
3	temp	4.50
12	month_5	3.43
11	month_4	3.10
8	winter	2.46
13	month_6	2.27
4	hum	1.98
2	workingday	1.89
14	month_8	1.83
17	sat	1.79
10	month_3	1.63
18	w_mist	1.58
16	month_10	1.54
15	month_9	1.46
19	w_light	1.27
5	windspeed	1.22
1	holiday	1.17
9	yr_2019	1.04

Insight:

1. VIF is high for fall when compare to others so drop and check

#dropping fall column
X_train_new = X_train_new.drop(['fall'], axis = 1)

Rebuilding model without fall -3

# add constant
X_train_new = sm.add_constant(X_train_new)

# Create fitting model 
lm = sm.OLS(y_train, X_train_new).fit()

lm.summary()

OLS Regression Results

Dep. Variable:	cnt	R-squared:	0.850
Model:	OLS	Adj. R-squared:	0.845
Method:	Least Squares	F-statistic:	155.1
Date:	Sun, 27 Aug 2023	Prob (F-statistic):	2.72e-189
Time:	16:05:18	Log-Likelihood:	523.03
No. Observations:	510	AIC:	-1008.
Df Residuals:	491	BIC:	-927.6
Df Model:	18
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
const	0.1902	0.029	6.534	0.000	0.133	0.247
holiday	-0.0523	0.027	-1.949	0.052	-0.105	0.000
workingday	0.0434	0.011	3.776	0.000	0.021	0.066
temp	0.5230	0.024	21.523	0.000	0.475	0.571
hum	-0.1692	0.037	-4.518	0.000	-0.243	-0.096
windspeed	-0.1894	0.025	-7.472	0.000	-0.239	-0.140
summer	0.0684	0.020	3.470	0.001	0.030	0.107
winter	0.1424	0.012	11.622	0.000	0.118	0.167
yr_2019	0.2293	0.008	28.788	0.000	0.214	0.245
month_3	0.0392	0.016	2.394	0.017	0.007	0.071
month_4	0.0435	0.026	1.699	0.090	-0.007	0.094
month_5	0.0584	0.026	2.268	0.024	0.008	0.109
month_6	0.0323	0.022	1.448	0.148	-0.012	0.076
month_8	0.0676	0.017	3.936	0.000	0.034	0.101
month_9	0.1344	0.017	8.047	0.000	0.102	0.167
month_10	0.0430	0.017	2.513	0.012	0.009	0.077
sat	0.0529	0.014	3.656	0.000	0.024	0.081
w_mist	-0.0584	0.010	-5.648	0.000	-0.079	-0.038
w_light	-0.2480	0.026	-9.540	0.000	-0.299	-0.197

Omnibus:	71.760	Durbin-Watson:	2.031
Prob(Omnibus):	0.000	Jarque-Bera (JB):	184.663
Skew:	-0.710	Prob(JB):	7.96e-41
Kurtosis:	5.583	Cond. No.	21.1

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

Insight

1. $ R^2 and Adj. R^2$ values are fine (0.85, 0.84) - no impact of dropping column fall

#create dataframe for all the features and their VIF's
vif = pd.DataFrame()
vif['Features'] = X_train_new.columns
vif['VIF'] = [variance_inflation_factor(X_train_new.values, i) for i in range(X_train_new.shape[1])]
vif['VIF'] = round(vif['VIF'],2)
vif = vif.sort_values(by = 'VIF' , ascending = False)
vif

	Features	VIF
0	const	55.28
6	summer	4.69
11	month_5	3.33
10	month_4	3.10
12	month_6	2.23
3	temp	1.96
4	hum	1.94
2	workingday	1.89
7	winter	1.83
16	sat	1.79
13	month_8	1.67
17	w_mist	1.57
9	month_3	1.54
15	month_10	1.51
14	month_9	1.34
18	w_light	1.26
5	windspeed	1.21
1	holiday	1.17
8	yr_2019	1.03

Insight:

1. VIF is more for summer when compare to others so drop and check

#dropping summer column
X_train_new = X_train_new.drop(['summer'], axis = 1)

Rebuilding model without summer -4

# add constant
X_train_new = sm.add_constant(X_train_new)

# Create fitting model 
lm = sm.OLS(y_train, X_train_new).fit()

print(lm.summary())

                            OLS Regression Results                            
==============================================================================
Dep. Variable:                    cnt   R-squared:                       0.847
Model:                            OLS   Adj. R-squared:                  0.841
Method:                 Least Squares   F-statistic:                     159.9
Date:                Sun, 27 Aug 2023   Prob (F-statistic):          7.83e-188
Time:                        16:05:18   Log-Likelihood:                 516.85
No. Observations:                 510   AIC:                            -997.7
Df Residuals:                     492   BIC:                            -921.5
Df Model:                          17                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
const          0.1856      0.029      6.313      0.000       0.128       0.243
holiday       -0.0515      0.027     -1.899      0.058      -0.105       0.002
workingday     0.0439      0.012      3.775      0.000       0.021       0.067
temp           0.5190      0.025     21.146      0.000       0.471       0.567
hum           -0.1602      0.038     -4.242      0.000      -0.234      -0.086
windspeed     -0.1892      0.026     -7.381      0.000      -0.240      -0.139
winter         0.1417      0.012     11.437      0.000       0.117       0.166
yr_2019        0.2293      0.008     28.475      0.000       0.213       0.245
month_3        0.0638      0.015      4.278      0.000       0.034       0.093
month_4        0.1125      0.016      6.871      0.000       0.080       0.145
month_5        0.1267      0.017      7.571      0.000       0.094       0.160
month_6        0.0771      0.018      4.198      0.000       0.041       0.113
month_8        0.0685      0.017      3.945      0.000       0.034       0.103
month_9        0.1345      0.017      7.967      0.000       0.101       0.168
month_10       0.0431      0.017      2.491      0.013       0.009       0.077
sat            0.0544      0.015      3.715      0.000       0.026       0.083
w_mist        -0.0577      0.010     -5.526      0.000      -0.078      -0.037
w_light       -0.2482      0.026     -9.444      0.000      -0.300      -0.197
==============================================================================
Omnibus:                       65.477   Durbin-Watson:                   2.002
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              172.023
Skew:                          -0.643   Prob(JB):                     4.42e-38
Kurtosis:                       5.538   Cond. No.                         20.8
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

Insight

1. $ R^2 and Adj. R^2$ values are fine (0.84)

#create dataframe for all the features and their VIF's
vif = pd.DataFrame()
vif['Features'] = X_train_new.columns
vif['VIF'] = [variance_inflation_factor(X_train_new.values, i) for i in range(X_train_new.shape[1])]
vif['VIF'] = round(vif['VIF'],2)
vif = vif.sort_values(by = 'VIF' , ascending = False)
vif

	Features	VIF
0	const	55.16
3	temp	1.96
4	hum	1.93
2	workingday	1.89
6	winter	1.83
15	sat	1.79
12	month_8	1.67
16	w_mist	1.57
14	month_10	1.51
11	month_6	1.48
10	month_5	1.38
13	month_9	1.34
17	w_light	1.26
8	month_3	1.25
9	month_4	1.24
5	windspeed	1.21
1	holiday	1.17
7	yr_2019	1.03

Insight:

1. For holiday p-value - 0.058 and VIF 1.17 , lets drop and check

#dropping holiday column
X_train_new = X_train_new.drop(['holiday'], axis = 1)

Rebuilding model without holiday -5

# add constant
X_train_new = sm.add_constant(X_train_new)

# Create fitting model 
lm = sm.OLS(y_train, X_train_new).fit()

print(lm.summary())

                            OLS Regression Results                            
==============================================================================
Dep. Variable:                    cnt   R-squared:                       0.846
Model:                            OLS   Adj. R-squared:                  0.841
Method:                 Least Squares   F-statistic:                     168.8
Date:                Sun, 27 Aug 2023   Prob (F-statistic):          3.54e-188
Time:                        16:05:18   Log-Likelihood:                 514.99
No. Observations:                 510   AIC:                            -996.0
Df Residuals:                     493   BIC:                            -924.0
Df Model:                          16                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
const          0.1765      0.029      6.067      0.000       0.119       0.234
workingday     0.0515      0.011      4.705      0.000       0.030       0.073
temp           0.5199      0.025     21.132      0.000       0.472       0.568
hum           -0.1592      0.038     -4.205      0.000      -0.234      -0.085
windspeed     -0.1900      0.026     -7.395      0.000      -0.240      -0.140
winter         0.1413      0.012     11.378      0.000       0.117       0.166
yr_2019        0.2295      0.008     28.424      0.000       0.214       0.245
month_3        0.0652      0.015      4.368      0.000       0.036       0.095
month_4        0.1132      0.016      6.896      0.000       0.081       0.145
month_5        0.1280      0.017      7.638      0.000       0.095       0.161
month_6        0.0786      0.018      4.275      0.000       0.042       0.115
month_8        0.0694      0.017      3.989      0.000       0.035       0.104
month_9        0.1333      0.017      7.882      0.000       0.100       0.167
month_10       0.0444      0.017      2.559      0.011       0.010       0.078
sat            0.0621      0.014      4.403      0.000       0.034       0.090
w_mist        -0.0575      0.010     -5.489      0.000      -0.078      -0.037
w_light       -0.2475      0.026     -9.394      0.000      -0.299      -0.196
==============================================================================
Omnibus:                       70.585   Durbin-Watson:                   1.981
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              193.116
Skew:                          -0.679   Prob(JB):                     1.16e-42
Kurtosis:                       5.692   Cond. No.                         20.7
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

Insight

1. $ R^2 and Adj. R^2$ values are fine (0.84)
2. P-values for all features are low

#create dataframe for all the features and their VIF's
vif = pd.DataFrame()
vif['Features'] = X_train_new.columns
vif['VIF'] = [variance_inflation_factor(X_train_new.values, i) for i in range(X_train_new.shape[1])]
vif['VIF'] = round(vif['VIF'],2)
vif = vif.sort_values(by = 'VIF' , ascending = False)
vif

	Features	VIF
0	const	53.68
2	temp	1.96
3	hum	1.93
5	winter	1.83
11	month_8	1.67
1	workingday	1.66
14	sat	1.65
15	w_mist	1.57
13	month_10	1.50
10	month_6	1.48
9	month_5	1.38
12	month_9	1.34
16	w_light	1.26
7	month_3	1.25
8	month_4	1.23
4	windspeed	1.21
6	yr_2019	1.03

Insight :

VIF values are also low for all the features

Let us try and check the impact of dropping temp variable

#dropping temp column 
X_train_new1 = X_train_new
X_train_new1 = X_train_new1.drop(['temp'], axis = 1)

# add constant
X_train_new1 = sm.add_constant(X_train_new1)

# Create fitting model 
lm1 = sm.OLS(y_train, X_train_new1).fit()

print(lm1.summary())

                            OLS Regression Results                            
==============================================================================
Dep. Variable:                    cnt   R-squared:                       0.706
Model:                            OLS   Adj. R-squared:                  0.697
Method:                 Least Squares   F-statistic:                     79.03
Date:                Sun, 27 Aug 2023   Prob (F-statistic):          1.21e-120
Time:                        16:05:18   Log-Likelihood:                 350.54
No. Observations:                 510   AIC:                            -669.1
Df Residuals:                     494   BIC:                            -601.3
Df Model:                          15                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
const          0.3186      0.039      8.164      0.000       0.242       0.395
workingday     0.0585      0.015      3.880      0.000       0.029       0.088
hum           -0.0013      0.051     -0.026      0.979      -0.102       0.099
windspeed     -0.2249      0.035     -6.361      0.000      -0.294      -0.155
winter         0.1028      0.017      6.066      0.000       0.069       0.136
yr_2019        0.2502      0.011     22.638      0.000       0.228       0.272
month_3        0.0597      0.021      2.900      0.004       0.019       0.100
month_4        0.1521      0.022      6.763      0.000       0.108       0.196
month_5        0.2294      0.022     10.354      0.000       0.186       0.273
month_6        0.2495      0.023     10.954      0.000       0.205       0.294
month_8        0.2488      0.021     11.877      0.000       0.208       0.290
month_9        0.2618      0.022     12.024      0.000       0.219       0.305
month_10       0.1170      0.023      4.992      0.000       0.071       0.163
sat            0.0635      0.019      3.268      0.001       0.025       0.102
w_mist        -0.1056      0.014     -7.493      0.000      -0.133      -0.078
w_light       -0.2905      0.036     -8.019      0.000      -0.362      -0.219
==============================================================================
Omnibus:                       21.888   Durbin-Watson:                   1.858
Prob(Omnibus):                  0.000   Jarque-Bera (JB):               53.367
Skew:                           0.131   Prob(JB):                     2.58e-12
Kurtosis:                       4.563   Cond. No.                         19.6
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

Insight:

After removing temp column, there is huge impact on the R-squared and Adj. R- value so revering the changes back and no more dropping of columns

Final Model

# Final Model
# add constant
X_train_new = sm.add_constant(X_train_new)

# Create fitting model 
lm = sm.OLS(y_train, X_train_new).fit()

print(lm.summary())

                            OLS Regression Results                            
==============================================================================
Dep. Variable:                    cnt   R-squared:                       0.846
Model:                            OLS   Adj. R-squared:                  0.841
Method:                 Least Squares   F-statistic:                     168.8
Date:                Sun, 27 Aug 2023   Prob (F-statistic):          3.54e-188
Time:                        16:05:18   Log-Likelihood:                 514.99
No. Observations:                 510   AIC:                            -996.0
Df Residuals:                     493   BIC:                            -924.0
Df Model:                          16                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
const          0.1765      0.029      6.067      0.000       0.119       0.234
workingday     0.0515      0.011      4.705      0.000       0.030       0.073
temp           0.5199      0.025     21.132      0.000       0.472       0.568
hum           -0.1592      0.038     -4.205      0.000      -0.234      -0.085
windspeed     -0.1900      0.026     -7.395      0.000      -0.240      -0.140
winter         0.1413      0.012     11.378      0.000       0.117       0.166
yr_2019        0.2295      0.008     28.424      0.000       0.214       0.245
month_3        0.0652      0.015      4.368      0.000       0.036       0.095
month_4        0.1132      0.016      6.896      0.000       0.081       0.145
month_5        0.1280      0.017      7.638      0.000       0.095       0.161
month_6        0.0786      0.018      4.275      0.000       0.042       0.115
month_8        0.0694      0.017      3.989      0.000       0.035       0.104
month_9        0.1333      0.017      7.882      0.000       0.100       0.167
month_10       0.0444      0.017      2.559      0.011       0.010       0.078
sat            0.0621      0.014      4.403      0.000       0.034       0.090
w_mist        -0.0575      0.010     -5.489      0.000      -0.078      -0.037
w_light       -0.2475      0.026     -9.394      0.000      -0.299      -0.196
==============================================================================
Omnibus:                       70.585   Durbin-Watson:                   1.981
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              193.116
Skew:                          -0.679   Prob(JB):                     1.16e-42
Kurtosis:                       5.692   Cond. No.                         20.7
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

Insights:

1.Final $ R^2 $ is 0.84 (decent) , Adj. $ R^2 $ is 0.84(decent), prob. F stats is less than 0.05 (overall model fit is significant) which is good fit

2.The Top 3 Features : temp, yr_2019, windspeed

(writting eqn only with top 3 - we need to include rest as well)

cnt = 0.1765 + (0.5199 * temp) + (0.2995 * yr_2019) - (0.19 * windspeed)......

Positive impact:

1. Temp
2. Year - yr_2019
3. months - 4(april),5(may),9(sept)

Negative impact:

1. Weather : weather light
2. windspeed
3. hum - humidity

Step 5: Residual Analysis of the train data

# Predict the value (y_train_cnt)
y_train_cnt = lm.predict(X_train_new)

# plot histogram for error term
fig = plt.figure()
sns.distplot((y_train-y_train_cnt), bins = 20)
fig.suptitle("Error Term", fontsize = 20)
plt.xlabel("error",fontsize = 10)

Text(0.5, 0, 'error')

Insight:

1. Histogram of the Error Term plot is normal distribution with mean value 0

Step 6: Making Predictions

col_list = X_train_new.columns
col_list

Index(['const', 'workingday', 'temp', 'hum', 'windspeed', 'winter', 'yr_2019',
       'month_3', 'month_4', 'month_5', 'month_6', 'month_8', 'month_9',
       'month_10', 'sat', 'w_mist', 'w_light'],
      dtype='object')

col_list = ['workingday', 'temp', 'hum', 'windspeed', 'winter', 'yr_2019',
       'month_3', 'month_4', 'month_5', 'month_6', 'month_8', 'month_9',
       'month_10', 'sat', 'w_mist', 'w_light']

# apply scalar for test data for numeric variables
df_test[num_vars] = scaler.transform(df_test[num_vars])

Dividing X and y variables

y_test = df_test.pop('cnt')

# maintain same columns as in final model dataset
X_test = df_test[col_list]

#add constant
X_test = sm.add_constant(X_test)

#Making Prediction
y_pred = lm.predict(X_test)

Step 7: Model Evaluation

# ploting y_test and y_pred (predicted values) to understand the spread
fig = plt.figure()
plt.scatter(y_test,y_pred)

#plot title
fig.suptitle("y_test and y_pred plot", fontsize = 20)

# X label
plt.xlabel('y_test', fontsize = 10)

#Y label
plt.ylabel('y_pred', fontsize = 10)

plt.show()

Insight:

Normal distribution evident

r2_Score

from sklearn.metrics import r2_score
r2_score(y_test,y_pred)

0.8186483411107541

Insight:

Model is fine/fit